`f''(x) = x^2, f'(0) = 8, f(0) = 4` Find the particular solution that satisfies the differential equation.

Expert Answers
gsarora17 eNotes educator| Certified Educator

`f''(x)=x^2`

`f'(x)=intx^2dx`

`f'(x)=x^3/3+C_1`

Now let's find C_1 , given f'(0)=8

`8=0^3/3+C_1`

`C_1=8`

`:.f'(x)=x^3/3+8`

`f(x)=int(x^3/3+8)dx`

`f(x)=(1/3)((x^(3+1))/(3+1))+8x+C_2`

`f(x)=x^4/12+8x+C_2`

Now let's find constant C_2 , given f(0)=4,

`4=0^4/12+8(0)+C_2`

`C_2=4`

`:.f(x)=x^4/12+8x+4`